Measure and Category pp 27-30 | Cite as

# The Banach-Mazur Game

## Abstract

Around 1928, the Polish mathematician S. Mazur invented the following mathematical “game.” Player (*A*) is “dealt” an arbitrary subset *A* of a closed interval *I*_{0}. The complementary set *B* = *I*_{0} − *A* is dealt to player (*B*). The game 〈*A, B*〉 is played as follows: (*A*) chooses arbitrarily a closed interval *I*_{1}⊂*I*_{0}; then (*B*) chooses a closed interval *I*_{2}⊂*I*_{1}; then (*A*) chooses a closed interval *I*_{3}⊂*I*_{2}; and so on, alternately. Together the players determine a nested sequence of closed intervals *I*_{n}, (*A*) choosing those with odd index, (*B*) those with even index. If the set ∩*I*_{ n } has at least one point in common with *A*, then (*A*) wins; otherwise, (*B*) wins.

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